# gravin-slides

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```					                                          General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Truthfulness and frugality ratio in the cheapest
path auctions
Estonian Theory Days 2009

Nick Gravin
(based on the unpublished joint paper with E. Elkind,
N. Chen,F. Petrov)

St.Petersburg Department of Steklov Mathematical Institute RAS,

Nanyang Technological University, Singapore

October 2009

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

General set system auctions. Model:

There are given a set of agents E = {1, 2, . . . , n} and a family
of feasible sets F ⊆ 2E .
Auctioneer is intent on hiring a team (some feasible set) of
agents — winning set.
Each agent i has a true cost ci , but at the auction he can bid
another price bi ≥ ci .
Utility ui (proﬁt) of an agent is bi − ci if he wins and 0 if he
loses.
Every agent is selﬁsh and wants to maximize ui .

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

General set system auctions. Model:

There are given a set of agents E = {1, 2, . . . , n} and a family
of feasible sets F ⊆ 2E .
Auctioneer is intent on hiring a team (some feasible set) of
agents — winning set.
Each agent i has a true cost ci , but at the auction he can bid
another price bi ≥ ci .
Utility ui (proﬁt) of an agent is bi − ci if he wins and 0 if he
loses.
Every agent is selﬁsh and wants to maximize ui .

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

General set system auctions. Model:

There are given a set of agents E = {1, 2, . . . , n} and a family
of feasible sets F ⊆ 2E .
Auctioneer is intent on hiring a team (some feasible set) of
agents — winning set.
Each agent i has a true cost ci , but at the auction he can bid
another price bi ≥ ci .
Utility ui (proﬁt) of an agent is bi − ci if he wins and 0 if he
loses.
Every agent is selﬁsh and wants to maximize ui .

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

General set system auctions. Model:

There are given a set of agents E = {1, 2, . . . , n} and a family
of feasible sets F ⊆ 2E .
Auctioneer is intent on hiring a team (some feasible set) of
agents — winning set.
Each agent i has a true cost ci , but at the auction he can bid
another price bi ≥ ci .
Utility ui (proﬁt) of an agent is bi − ci if he wins and 0 if he
loses.
Every agent is selﬁsh and wants to maximize ui .

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

General set system auctions. Model:

There are given a set of agents E = {1, 2, . . . , n} and a family
of feasible sets F ⊆ 2E .
Auctioneer is intent on hiring a team (some feasible set) of
agents — winning set.
Each agent i has a true cost ci , but at the auction he can bid
another price bi ≥ ci .
Utility ui (proﬁt) of an agent is bi − ci if he wins and 0 if he
loses.
Every agent is selﬁsh and wants to maximize ui .

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

First price rule

Auction is called a ﬁrst price if auctioneer always chooses the
cheapest feasible set, i.e. a set F with minimal e∈F b(e).

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

examples

Non-trivial issues concerning computational complexity appears
when feasible sets are given implicitly.
1    Path auction: agents — edges in the graph; feasible sets —
all paths between two given vertices s and t.
2    k-Path: agents — edges in the graph; feasible sets — k
edge-disjoint paths from s to t.
3    Vertex cover auction: agents — vertices; feasible sets — all
vertex covers of edges.
4    Spanning tree auction: agents — edges; feasible sets —
spanning trees.
5    Perfect matching auction: agents — edges in bipartite
graph; feasible sets — perfect matchings.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

examples

Non-trivial issues concerning computational complexity appears
when feasible sets are given implicitly.
1    Path auction: agents — edges in the graph; feasible sets —
all paths between two given vertices s and t.
2    k-Path: agents — edges in the graph; feasible sets — k
edge-disjoint paths from s to t.
3    Vertex cover auction: agents — vertices; feasible sets — all
vertex covers of edges.
4    Spanning tree auction: agents — edges; feasible sets —
spanning trees.
5    Perfect matching auction: agents — edges in bipartite
graph; feasible sets — perfect matchings.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

examples

Non-trivial issues concerning computational complexity appears
when feasible sets are given implicitly.
1    Path auction: agents — edges in the graph; feasible sets —
all paths between two given vertices s and t.
2    k-Path: agents — edges in the graph; feasible sets — k
edge-disjoint paths from s to t.
3    Vertex cover auction: agents — vertices; feasible sets — all
vertex covers of edges.
4    Spanning tree auction: agents — edges; feasible sets —
spanning trees.
5    Perfect matching auction: agents — edges in bipartite
graph; feasible sets — perfect matchings.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

examples

Non-trivial issues concerning computational complexity appears
when feasible sets are given implicitly.
1    Path auction: agents — edges in the graph; feasible sets —
all paths between two given vertices s and t.
2    k-Path: agents — edges in the graph; feasible sets — k
edge-disjoint paths from s to t.
3    Vertex cover auction: agents — vertices; feasible sets — all
vertex covers of edges.
4    Spanning tree auction: agents — edges; feasible sets —
spanning trees.
5    Perfect matching auction: agents — edges in bipartite
graph; feasible sets — perfect matchings.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

examples

Non-trivial issues concerning computational complexity appears
when feasible sets are given implicitly.
1    Path auction: agents — edges in the graph; feasible sets —
all paths between two given vertices s and t.
2    k-Path: agents — edges in the graph; feasible sets — k
edge-disjoint paths from s to t.
3    Vertex cover auction: agents — vertices; feasible sets — all
vertex covers of edges.
4    Spanning tree auction: agents — edges; feasible sets —
spanning trees.
5    Perfect matching auction: agents — edges in bipartite
graph; feasible sets — perfect matchings.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

examples

Non-trivial issues concerning computational complexity appears
when feasible sets are given implicitly.
1    Path auction: agents — edges in the graph; feasible sets —
all paths between two given vertices s and t.
2    k-Path: agents — edges in the graph; feasible sets — k
edge-disjoint paths from s to t.
3    Vertex cover auction: agents — vertices; feasible sets — all
vertex covers of edges.
4    Spanning tree auction: agents — edges; feasible sets —
spanning trees.
5    Perfect matching auction: agents — edges in bipartite
graph; feasible sets — perfect matchings.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Nash equilibrium in ﬁrst price auction

Deﬁnition
Nash equilibrium(NE) in the ﬁrst price auction is an assignment of
agent bids b, such that no agent e can increase its utility ue by
varying its bid be .

NE exists if there is no agent belonging to all feasible
sets(monopolist).
We can ﬁnd a NE explicitly in polynomial time.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Nash equilibrium in ﬁrst price auction

Deﬁnition
Nash equilibrium(NE) in the ﬁrst price auction is an assignment of
agent bids b, such that no agent e can increase its utility ue by
varying its bid be .

NE exists if there is no agent belonging to all feasible
sets(monopolist).
We can ﬁnd a NE explicitly in polynomial time.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Nash equilibrium in ﬁrst price auction

Deﬁnition
Nash equilibrium(NE) in the ﬁrst price auction is an assignment of
agent bids b, such that no agent e can increase its utility ue by
varying its bid be .

NE exists if there is no agent belonging to all feasible
sets(monopolist).
We can ﬁnd a NE explicitly in polynomial time.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Truthful mechanisms

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Mechanisms for winners and payments selection

Suppose the true costs of all agents are private.
Agents have only one chance to send their bids to the
auctioneer.
Auctioneer (buyer, center) on the base of these bids should
choose the winning set and how much to pay to them.

Deﬁnition
The mechanism of winners selection and their payments is called
truthful if each agent maximizes utility by bidding its true cost
(for any ﬁxed bids of other agents).

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Mechanisms for winners and payments selection

Suppose the true costs of all agents are private.
Agents have only one chance to send their bids to the
auctioneer.
Auctioneer (buyer, center) on the base of these bids should
choose the winning set and how much to pay to them.

Deﬁnition
The mechanism of winners selection and their payments is called
truthful if each agent maximizes utility by bidding its true cost
(for any ﬁxed bids of other agents).

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Mechanisms for winners and payments selection

Suppose the true costs of all agents are private.
Agents have only one chance to send their bids to the
auctioneer.
Auctioneer (buyer, center) on the base of these bids should
choose the winning set and how much to pay to them.

Deﬁnition
The mechanism of winners selection and their payments is called
truthful if each agent maximizes utility by bidding its true cost
(for any ﬁxed bids of other agents).

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Mechanisms for winners and payments selection

Suppose the true costs of all agents are private.
Agents have only one chance to send their bids to the
auctioneer.
Auctioneer (buyer, center) on the base of these bids should
choose the winning set and how much to pay to them.

Deﬁnition
The mechanism of winners selection and their payments is called
truthful if each agent maximizes utility by bidding its true cost
(for any ﬁxed bids of other agents).

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Example

(Vickrey) Second price auction (for particular set system).
There are n sellers and auctioneer needs to bay exactly one thing.
Mechanism selects the cheapest agent and pays to him the bid of
the second by the price agent.

Is it clear why this mechanism is truthful?

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Example

(Vickrey) Second price auction (for particular set system).
There are n sellers and auctioneer needs to bay exactly one thing.
Mechanism selects the cheapest agent and pays to him the bid of
the second by the price agent.

Is it clear why this mechanism is truthful?

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Example

(Vickrey-Clarke-Groves) VCG for general set system.
Choose the cheapest feasible set F and pay independently to
each agent in F its threshold bid, i.e. such bid that F is still the
cheapest feasible set.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms               √
KKT       -mechanism
k-paths and cheapest path auctions

VCG for cheapest path

c:                                   0                0
0
0                                                         0
S                                   1                                  T

Initial true costs.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms               √
KKT       -mechanism
k-paths and cheapest path auctions

VCG for cheapest path

c:                                   0
0                             0
0                                                       0
S                                   1                                  T

Cheapest path.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms                √
KKT       -mechanism
k-paths and cheapest path auctions

VCG for cheapest path

b:                                    1
5
1                               1
5                             5
1                                                            1
5                                                    5
S                                   1                                    T

Cheapest Nash equilibrium. Total payment is 1.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms               √
KKT       -mechanism
k-paths and cheapest path auctions

VCG for cheapest path

c:                                   0
1                             0
0                                                       0
S                                   1                                  T

Payment to each agent on the winning path should be 1.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms               √
KKT       -mechanism
k-paths and cheapest path auctions

VCG for cheapest path

c:                                   1
1                             1
1                                                       1
S                                   1                                  T

The total payment is 5, but for NE it was only 1.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

How much mechanism overpays

(E, F) — monopoly free set system;
M — truthful mechanism for (E, F).
c is a true cost vector; ν(c) — the total payment in the
cheapest Nash for (E, F, c);
PM (c) — total payment of M for c;

Deﬁnition
Frugality ratio(Karlin, Kempe, Tamir FOCS 05)measures a
mechanism eﬃciency.

PM (c)
ΦM = sup                  .
c    ν(c)

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

How much mechanism overpays

(E, F) — monopoly free set system;
M — truthful mechanism for (E, F).
c is a true cost vector; ν(c) — the total payment in the
cheapest Nash for (E, F, c);
PM (c) — total payment of M for c;

Deﬁnition
Frugality ratio(Karlin, Kempe, Tamir FOCS 05)measures a
mechanism eﬃciency.

PM (c)
ΦM = sup                  .
c    ν(c)

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

How much mechanism overpays

(E, F) — monopoly free set system;
M — truthful mechanism for (E, F).
c is a true cost vector; ν(c) — the total payment in the
cheapest Nash for (E, F, c);
PM (c) — total payment of M for c;

Deﬁnition
Frugality ratio(Karlin, Kempe, Tamir FOCS 05)measures a
mechanism eﬃciency.

PM (c)
ΦM = sup                  .
c    ν(c)

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

How much mechanism overpays

(E, F) — monopoly free set system;
M — truthful mechanism for (E, F).
c is a true cost vector; ν(c) — the total payment in the
cheapest Nash for (E, F, c);
PM (c) — total payment of M for c;

Deﬁnition
Frugality ratio(Karlin, Kempe, Tamir FOCS 05)measures a
mechanism eﬃciency.

PM (c)
ΦM = sup                  .
c    ν(c)

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

How much mechanism overpays

(E, F) — monopoly free set system;
M — truthful mechanism for (E, F).
c is a true cost vector; ν(c) — the total payment in the
cheapest Nash for (E, F, c);
PM (c) — total payment of M for c;

Deﬁnition
Frugality ratio(Karlin, Kempe, Tamir FOCS 05)measures a
mechanism eﬃciency.

PM (c)
ΦM = sup                  .
c    ν(c)

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Motivation for frugality ratio

ΦM ≥ 1, since one can take c to be by itself a Nash
equilibrium and then PM (c) ≥ ν(c).
ΦM = 1 ⇔ (E, F) is a matroid.(KKT 05 )
(E, F) is a Matroid iﬀ for every two sets S, T ∈ F, there is a
bijection f between S \ T and T \ S such that
S \ {e} ∪ {f (e)} is in F for every e ∈ S \ T .

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Motivation for frugality ratio

ΦM ≥ 1, since one can take c to be by itself a Nash
equilibrium and then PM (c) ≥ ν(c).
ΦM = 1 ⇔ (E, F) is a matroid.(KKT 05 )
(E, F) is a Matroid iﬀ for every two sets S, T ∈ F, there is a
bijection f between S \ T and T \ S such that
S \ {e} ∪ {f (e)} is in F for every e ∈ S \ T .

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Motivation for frugality ratio

ΦM ≥ 1, since one can take c to be by itself a Nash
equilibrium and then PM (c) ≥ ν(c).
ΦM = 1 ⇔ (E, F) is a matroid.(KKT 05 )
(E, F) is a Matroid iﬀ for every two sets S, T ∈ F, there is a
bijection f between S \ T and T \ S such that
S \ {e} ∪ {f (e)} is in F for every e ∈ S \ T .

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Mechanism design for cheapest path auctions

Known results:
VCG — ΦVCG for example with long path and an edge is
quadratic in terms of the optimal frugality ratio.
√
KKT      -mechanism gives a linear approximation to the
optimal frugality ratio (Φ√ ≤ 2X and any truthful
1
mechanism has frugality ratio at least √2 X ).
Pruning-lifting mechanism (our paper) for k-path auctions
provides optimal frugality ratio.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Mechanism design for cheapest path auctions

Known results:
VCG — ΦVCG for example with long path and an edge is
quadratic in terms of the optimal frugality ratio.
√
KKT      -mechanism gives a linear approximation to the
optimal frugality ratio (Φ√ ≤ 2X and any truthful
1
mechanism has frugality ratio at least √2 X ).
Pruning-lifting mechanism (our paper) for k-path auctions
provides optimal frugality ratio.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Mechanism design for cheapest path auctions

Known results:
VCG — ΦVCG for example with long path and an edge is
quadratic in terms of the optimal frugality ratio.
√
KKT      -mechanism gives a linear approximation to the
optimal frugality ratio (Φ√ ≤ 2X and any truthful
1
mechanism has frugality ratio at least √2 X ).
Pruning-lifting mechanism (our paper) for k-path auctions
provides optimal frugality ratio.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

KKT mechanism for cheapest path

√
-mechanism for cheapest path auctions.

We will see that for some graphs it will have much better frugality
ratio than VCG.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

First step.
Find two edge-disjoint paths P, P minimizing b(P) + b(P ).
(Ignore the rest of the graph)

S                                                                           T

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Second step.
Let s = v1 , v2 , . . . , vk+1 = t be the vertices that P, P have in
common, in the order in which they appear in P and P . Let Pi
(resp. Pi ) be the subpath of P (resp. P ) from vi to vi+1 .

v1                                         v2
v2                                 v3

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Third step.
For each i, include Pi in the solution iﬀ                              |Pi |b(Pi ) ≤           |Pi |b(Pi );
otherwise, include Pi .

b:                                                                 0
0                      3
v1                     5                    v2
v2                                 v3
1                                 1
0             4
1

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Forth step.
Pay to each winner its threshold bid (i.e. the largest value that he
can bid and still win, if the others bid the same).

b:                                                           4       2
−3

3
3

3 −
√

4
2
v1                   3 3                    v2

2
3
4
v2                                 v3
1                                 1
0             4
1

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Bounds on Φ√

√
-mechanism is truthful, thus b = c.
Φ√ ≤ 2 maxi                |Pi ||Pi |.
√
Thus for the edge and a path graph we have Φ√ = O(2 l),
while ΦVCG = O(l).
1
There is a lower bound                 √
2
maxi       |Pi ||Pi | on any truthful
mechanism, thus
1
√ max             |Pi ||Pi | ≤ Φ√ ≤ 2 max                     |Pi ||Pi |.
2 i                                                i

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Bounds on Φ√

√
-mechanism is truthful, thus b = c.
Φ√ ≤ 2 maxi                |Pi ||Pi |.
√
Thus for the edge and a path graph we have Φ√ = O(2 l),
while ΦVCG = O(l).
1
There is a lower bound                 √
2
maxi       |Pi ||Pi | on any truthful
mechanism, thus
1
√ max             |Pi ||Pi | ≤ Φ√ ≤ 2 max                     |Pi ||Pi |.
2 i                                                i

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Bounds on Φ√

√
-mechanism is truthful, thus b = c.
Φ√ ≤ 2 maxi                |Pi ||Pi |.
√
Thus for the edge and a path graph we have Φ√ = O(2 l),
while ΦVCG = O(l).
1
There is a lower bound                 √
2
maxi       |Pi ||Pi | on any truthful
mechanism, thus
1
√ max             |Pi ||Pi | ≤ Φ√ ≤ 2 max                     |Pi ||Pi |.
2 i                                                i

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

Bounds on Φ√

√
-mechanism is truthful, thus b = c.
Φ√ ≤ 2 maxi                |Pi ||Pi |.
√
Thus for the edge and a path graph we have Φ√ = O(2 l),
while ΦVCG = O(l).
1
There is a lower bound                 √
2
maxi       |Pi ||Pi | on any truthful
mechanism, thus
1
√ max             |Pi ||Pi | ≤ Φ√ ≤ 2 max                     |Pi ||Pi |.
2 i                                                i

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
VCG
Nash equilibriums
frugality ratio
Truthful mechanisms              √
KKT       -mechanism
k-paths and cheapest path auctions

k-paths and cheapest path auctions

k-paths and cheapest path auctions

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Nash for k-paths

Theorem
Any Nash equilibrium for the cheapest path auction with respect
to the bid vector b should have two non-intersecting by the edges
cheapest paths.

Theorem
Any Nash equilibrium for the k-paths auction with respect to the
bid vector b should have k + 1 non-intersecting by the edges
cheapest paths.

These theorems are non trivial results of graph theory.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Nash for k-paths

Theorem
Any Nash equilibrium for the cheapest path auction with respect
to the bid vector b should have two non-intersecting by the edges
cheapest paths.

Theorem
Any Nash equilibrium for the k-paths auction with respect to the
bid vector b should have k + 1 non-intersecting by the edges
cheapest paths.

These theorems are non trivial results of graph theory.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Nash for k-paths

Theorem
Any Nash equilibrium for the cheapest path auction with respect
to the bid vector b should have two non-intersecting by the edges
cheapest paths.

Theorem
Any Nash equilibrium for the k-paths auction with respect to the
bid vector b should have k + 1 non-intersecting by the edges
cheapest paths.

These theorems are non trivial results of graph theory.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Our mechanism for k-path auctions

Pruning-lifting mechanism

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Pruning step

Pick k + 1 edge-disjoint paths P1 , . . . , Pk+1 in G such that

δ(P1 , . . . , Pk+1 ) = max b(P)
P∈∪Pi

is minimized. (Ignore the rest of the graph.)
It is NP hard to ﬁnd P1 , . . . , Pk+1 even with < k + 1 factor
approximation of δ(P1 , . . . , Pk+1 ) to the optimal one.
The cheapest k + 1 ﬂow gives k + 1 approximation.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Pruning step

Pick k + 1 edge-disjoint paths P1 , . . . , Pk+1 in G such that

δ(P1 , . . . , Pk+1 ) = max b(P)
P∈∪Pi

is minimized. (Ignore the rest of the graph.)
It is NP hard to ﬁnd P1 , . . . , Pk+1 even with < k + 1 factor
approximation of δ(P1 , . . . , Pk+1 ) to the optimal one.
The cheapest k + 1 ﬂow gives k + 1 approximation.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Pruning step

Pick k + 1 edge-disjoint paths P1 , . . . , Pk+1 in G such that

δ(P1 , . . . , Pk+1 ) = max b(P)
P∈∪Pi

is minimized. (Ignore the rest of the graph.)
It is NP hard to ﬁnd P1 , . . . , Pk+1 even with < k + 1 factor
approximation of δ(P1 , . . . , Pk+1 ) to the optimal one.
The cheapest k + 1 ﬂow gives k + 1 approximation.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Lifting step

We construct a graph H as follows:
we take arcs of ∪i Pi as vertices for H;
we draw an edge between e and e iﬀ there is no path from s
to t containing both e, e .

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Lifting step

We construct a graph H as follows:
we take arcs of ∪i Pi as vertices for H;
we draw an edge between e and e iﬀ there is no path from s
to t containing both e, e .

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Lifting step

We construct a graph H as follows:
we take arcs of ∪i Pi as vertices for H;
we draw an edge between e and e iﬀ there is no path from s
to t containing both e, e .

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Lifting step

∪Pi

s                                                                                                         t

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Lifting step

∪Pi
11
00
00
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11                                                               00
11
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00

11
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00
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11                                                                           11
00
00
11                                                                           00
11
11
00

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Lifting step

H
1111111111111
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0000000000000
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00                                                               00
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0000000000000
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111
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1111111111111111111111
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11
00
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1111111111111111111111
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1111111111111111111111
111111
00000000
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111111
1111111111111111111111
00000000
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0000000000000
1111111111111                  000
11100
11                                           0
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00000000000
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1111111111111111111111
00000000
11111111
0000000000000
1111111111111                  111
00000
11                                           0
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00
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111111
00000000
11111111
1111111111111
0000000000000                  111
00000
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00000000000
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00          000000
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1111111111111111111111
00000000
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11
00
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11

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Weighting step

We split H into its components of connectivity H1 , . . . , Hl .
Let Ai be an adjacency matrix of Hi .
For each i we ﬁnd the positive eigenvector (wi ) and
eigenvalue αi of the matrix Ai . Let α = maxi αi .

(Ai ) (wi ) = αi (wi )
b(e)
Deﬁne b (e) = w (e) . Let P the maximum weight path with
respec to b (). Take ∪Pi \ P as the set of winners.
Pay to each agent its threshold bid.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Weighting step

We split H into its components of connectivity H1 , . . . , Hl .
Let Ai be an adjacency matrix of Hi .
For each i we ﬁnd the positive eigenvector (wi ) and
eigenvalue αi of the matrix Ai . Let α = maxi αi .

(Ai ) (wi ) = αi (wi )
b(e)
Deﬁne b (e) = w (e) . Let P the maximum weight path with
respec to b (). Take ∪Pi \ P as the set of winners.
Pay to each agent its threshold bid.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Weighting step

We split H into its components of connectivity H1 , . . . , Hl .
Let Ai be an adjacency matrix of Hi .
For each i we ﬁnd the positive eigenvector (wi ) and
eigenvalue αi of the matrix Ai . Let α = maxi αi .

(Ai ) (wi ) = αi (wi )
b(e)
Deﬁne b (e) = w (e) . Let P the maximum weight path with
respec to b (). Take ∪Pi \ P as the set of winners.
Pay to each agent its threshold bid.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

Weighting step

We split H into its components of connectivity H1 , . . . , Hl .
Let Ai be an adjacency matrix of Hi .
For each i we ﬁnd the positive eigenvector (wi ) and
eigenvalue αi of the matrix Ai . Let α = maxi αi .

(Ai ) (wi ) = αi (wi )
b(e)
Deﬁne b (e) = w (e) . Let P the maximum weight path with
respec to b (). Take ∪Pi \ P as the set of winners.
Pay to each agent its threshold bid.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

ΦPL

α
,         ΦPL =
k
where α is a lower bound on the frugality ratio of any truthful
k
mechanism.

So PL is the optimal mechanism with respect to frugality ratio.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)
General auctions
Nash equilibriums
Truthful mechanisms
k-paths and cheapest path auctions

ΦPL

α
,         ΦPL =
k
where α is a lower bound on the frugality ratio of any truthful
k
mechanism.

So PL is the optimal mechanism with respect to frugality ratio.

Truthfulness and frugality ratio in the cheapest path auctions
Nick Gravin(based on the unpublished joint paper with E. Elkind, N. Chen,F. Petrov)

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